Question: Find $(-2)^{3}+(-2)^{2}+(-2)^{1}+2^{1}+2^{2}+2^{3}$.
Solution: Recall that $(-a)^{n}=a^n$ for even $n$ and $(-a)^{n}=-a^n$ for odd $n$. This means that $(-2)^{3}=-2^{3}$, $(-2)^2=2^2$, and $(-2)^1=-2^1.$ Additionally, any number added to its opposite is equal to zero, so $-2^3+2^3=0$. Now we can use the commutative property of addition to obtain  \begin{align*}
(-2)^{3}+(-2)^{2}&+(-2)^{1}+2^{1}+2^{2}+2^{3}\\
&=-2^{3}+2^{2}+(-2^{1})+2^{1}+2^{2}+2^{3}\\
&=-2^{3}+2^{3}+(-2^{1})+2^{1}+2^{2}+2^{2}\\
&=0+2^{2}+2^{2}\\
&=4+4\\
&=\boxed{8}.
\end{align*}